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Theorem onintexmid 4350
Description: If the intersection (infimum) of an inhabited class of ordinal numbers belongs to the class, excluded middle follows. The hypothesis would be provable given excluded middle. (Contributed by Mario Carneiro and Jim Kingdon, 29-Aug-2021.)
Hypothesis
Ref Expression
onintexmid.onint ((𝑦 ⊆ On ∧ ∃𝑥 𝑥𝑦) → 𝑦𝑦)
Assertion
Ref Expression
onintexmid (𝜑 ∨ ¬ 𝜑)
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem onintexmid
Dummy variables 𝑢 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 prssi 3569 . . . . . 6 ((𝑢 ∈ On ∧ 𝑣 ∈ On) → {𝑢, 𝑣} ⊆ On)
2 prmg 3535 . . . . . . 7 (𝑢 ∈ On → ∃𝑥 𝑥 ∈ {𝑢, 𝑣})
32adantr 270 . . . . . 6 ((𝑢 ∈ On ∧ 𝑣 ∈ On) → ∃𝑥 𝑥 ∈ {𝑢, 𝑣})
4 zfpair2 4000 . . . . . . 7 {𝑢, 𝑣} ∈ V
5 sseq1 3031 . . . . . . . . 9 (𝑦 = {𝑢, 𝑣} → (𝑦 ⊆ On ↔ {𝑢, 𝑣} ⊆ On))
6 eleq2 2146 . . . . . . . . . 10 (𝑦 = {𝑢, 𝑣} → (𝑥𝑦𝑥 ∈ {𝑢, 𝑣}))
76exbidv 1748 . . . . . . . . 9 (𝑦 = {𝑢, 𝑣} → (∃𝑥 𝑥𝑦 ↔ ∃𝑥 𝑥 ∈ {𝑢, 𝑣}))
85, 7anbi12d 457 . . . . . . . 8 (𝑦 = {𝑢, 𝑣} → ((𝑦 ⊆ On ∧ ∃𝑥 𝑥𝑦) ↔ ({𝑢, 𝑣} ⊆ On ∧ ∃𝑥 𝑥 ∈ {𝑢, 𝑣})))
9 inteq 3665 . . . . . . . . 9 (𝑦 = {𝑢, 𝑣} → 𝑦 = {𝑢, 𝑣})
10 id 19 . . . . . . . . 9 (𝑦 = {𝑢, 𝑣} → 𝑦 = {𝑢, 𝑣})
119, 10eleq12d 2153 . . . . . . . 8 (𝑦 = {𝑢, 𝑣} → ( 𝑦𝑦 {𝑢, 𝑣} ∈ {𝑢, 𝑣}))
128, 11imbi12d 232 . . . . . . 7 (𝑦 = {𝑢, 𝑣} → (((𝑦 ⊆ On ∧ ∃𝑥 𝑥𝑦) → 𝑦𝑦) ↔ (({𝑢, 𝑣} ⊆ On ∧ ∃𝑥 𝑥 ∈ {𝑢, 𝑣}) → {𝑢, 𝑣} ∈ {𝑢, 𝑣})))
13 onintexmid.onint . . . . . . 7 ((𝑦 ⊆ On ∧ ∃𝑥 𝑥𝑦) → 𝑦𝑦)
144, 12, 13vtocl 2664 . . . . . 6 (({𝑢, 𝑣} ⊆ On ∧ ∃𝑥 𝑥 ∈ {𝑢, 𝑣}) → {𝑢, 𝑣} ∈ {𝑢, 𝑣})
151, 3, 14syl2anc 403 . . . . 5 ((𝑢 ∈ On ∧ 𝑣 ∈ On) → {𝑢, 𝑣} ∈ {𝑢, 𝑣})
16 elpri 3445 . . . . 5 ( {𝑢, 𝑣} ∈ {𝑢, 𝑣} → ( {𝑢, 𝑣} = 𝑢 {𝑢, 𝑣} = 𝑣))
1715, 16syl 14 . . . 4 ((𝑢 ∈ On ∧ 𝑣 ∈ On) → ( {𝑢, 𝑣} = 𝑢 {𝑢, 𝑣} = 𝑣))
18 incom 3176 . . . . . . 7 (𝑣𝑢) = (𝑢𝑣)
1918eqeq1i 2090 . . . . . 6 ((𝑣𝑢) = 𝑢 ↔ (𝑢𝑣) = 𝑢)
20 dfss1 3188 . . . . . 6 (𝑢𝑣 ↔ (𝑣𝑢) = 𝑢)
21 vex 2615 . . . . . . . 8 𝑢 ∈ V
22 vex 2615 . . . . . . . 8 𝑣 ∈ V
2321, 22intpr 3694 . . . . . . 7 {𝑢, 𝑣} = (𝑢𝑣)
2423eqeq1i 2090 . . . . . 6 ( {𝑢, 𝑣} = 𝑢 ↔ (𝑢𝑣) = 𝑢)
2519, 20, 243bitr4ri 211 . . . . 5 ( {𝑢, 𝑣} = 𝑢𝑢𝑣)
2623eqeq1i 2090 . . . . . 6 ( {𝑢, 𝑣} = 𝑣 ↔ (𝑢𝑣) = 𝑣)
27 dfss1 3188 . . . . . 6 (𝑣𝑢 ↔ (𝑢𝑣) = 𝑣)
2826, 27bitr4i 185 . . . . 5 ( {𝑢, 𝑣} = 𝑣𝑣𝑢)
2925, 28orbi12i 714 . . . 4 (( {𝑢, 𝑣} = 𝑢 {𝑢, 𝑣} = 𝑣) ↔ (𝑢𝑣𝑣𝑢))
3017, 29sylib 120 . . 3 ((𝑢 ∈ On ∧ 𝑣 ∈ On) → (𝑢𝑣𝑣𝑢))
3130rgen2a 2423 . 2 𝑢 ∈ On ∀𝑣 ∈ On (𝑢𝑣𝑣𝑢)
3231ordtri2or2exmid 4349 1 (𝜑 ∨ ¬ 𝜑)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 102  wo 662   = wceq 1285  wex 1422  wcel 1434  cin 2983  wss 2984  {cpr 3423   cint 3662  Oncon0 4153
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3922  ax-nul 3930  ax-pow 3974  ax-pr 3999  ax-un 4223  ax-setind 4315
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-ral 2358  df-rex 2359  df-rab 2362  df-v 2614  df-dif 2986  df-un 2988  df-in 2990  df-ss 2997  df-nul 3270  df-pw 3408  df-sn 3428  df-pr 3429  df-uni 3628  df-int 3663  df-tr 3902  df-iord 4156  df-on 4158  df-suc 4161
This theorem is referenced by: (None)
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